Fractional and Negative Indices: Fluency and Exam Drill - Worksheets, Questions and Revision

20 original exam-style questions - 7 pages of questions with a full mark scheme - free printable PDF.

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6.2D Fractional and Negative Indices: Fluency and Exam Drill

EDEXCEL 1MA1 · Calculators not allowed · about 90 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Key Index Laws to Recall

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Before you start: a^0 = 1 for any non-zero a. A negative index flips the base into a fraction: a^-n = 1/a^n. A fractional index is a root: a^(1/n) = nth root of a, and a^(m/n) = (nth root of a)^m. When multiplying powers of the same base, add the indices: a^m * a^n = a^(m+n). When dividing powers of the same base, subtract the indices: a^m / a^n = a^(m-n). When raising a power to a power, multiply the indices: (a^m)^n = a^(m*n). These rules still apply when m and n are fractions or negative numbers.

Work out the value of each expression. Give each answer as a fraction.

(a)4^-1(1)

(b)10^-2(1)

(Total for Question 1 is 3 marks)

Work out the value of each expression.

(a)25^1/2(1)

(b)64^1/3(1)

(Total for Question 2 is 3 marks)

Work out the value of each expression.

(a)8^2/3(2)

(b)16^3/4(2)

(Total for Question 3 is 6 marks)

Work out the value of each expression. Give each answer as a fraction.

(a)4^-1/2(2)

(b)27^-2/3(2)

(Total for Question 4 is 6 marks)

Simplify each expression. Give each answer as a single power of x.

(a)x^1/2 * x^3/2(2)

(b)x⁵ / x^-2(2)

(c)(x^2/3)³(2)

(Total for Question 5 is 6 marks)

Simplify each expression fully.

(a)(8x⁶)^1/3(2)

(b)(9x⁴)^1/2(2)

(c)(x³)^-2 * x⁴(2)

(Total for Question 6 is 6 marks)

Solve the equations:

(a)2^x = 32(1)

(b)3^x = 1/9(2)

(Total for Question 7 is 5 marks)

Solve the equations. Give each answer as a fraction.

(a)4^x = 8(2)

(b)9^x = 1/3(2)

(Total for Question 8 is 4 marks)

Simplify (x⁴)^1/2 / x^-3. Give your answer as a single power of x.

(Total for Question 9 is 3 marks)

Show that 27^1/3 * 4^-1/2 = 3/2

(Total for Question 10 is 3 marks)

Simplify √x³ / x^1/4. Give your answer as a single power of x.

(Total for Question 11 is 3 marks)

Solve 25^x = 5^3x-2

(Total for Question 12 is 3 marks)

Solve (1/8)^x = 16

(Total for Question 13 is 3 marks)

A population of bacteria is modelled by N = 800 * 4^-t/2, where N is the number of bacteria and t is the time in hours after the population is first measured.

(a)Work out N when t = 1(2)

(b)Find the value of t for which N = 100(3)

(Total for Question 14 is 5 marks)

Solve 8^x+1 = 4^2x-1

(Total for Question 15 is 4 marks)

Simplify (16x⁸)^3/4 fully.

(Total for Question 16 is 3 marks)

Simplify (x^-2/3)^-3. Give your answer as a single power of x.

(Total for Question 17 is 2 marks)

Given that 2^a = 8 and 2^b = 32, find the value of 2^a-b. Give your answer as a fraction.

(Total for Question 18 is 3 marks)

Solve the equation 9^x - 3^x+1 = 0

(Total for Question 19 is 4 marks)

Simplify (2ⁿ)^1/2 * 2^1-n fully. Give your answer in the form 2^k, where k is an expression in n.

(Total for Question 20 is 3 marks)

Mark scheme · 6.2D Fractional and Negative Indices: Fluency and Exam Drill

Question 1

(a) B1 1/4 cao
(a) Answer: 1/4
(b) B1 1/100 oe
(b) Answer: 1/100
(c) B1 1/16 cao
(c) Answer: 1/16

Question 2

(a) B1 5 cao
(a) Answer: 5
(b) B1 4 cao
(b) Answer: 4
(c) B1 2 cao
(c) Answer: 2

Question 3

(a) M1 8^1/3 = 2 found
(a) A1 4 cao
(a) Answer: 4
(b) M1 16^1/4 = 2 found
(b) A1 8 cao
(b) Answer: 8
(c) M1 25^1/2 = 5 found
(c) A1 125 cao
(c) Answer: 125

Question 4

(a) M1 4^1/2 = 2 found (or equivalent unit-fraction set up)
(a) A1 1/2 cao
(a) Answer: 1/2
(b) M1 27^2/3 = 9 found
(b) A1 1/9 cao
(b) Answer: 1/9
(c) M1 16^3/4 = 8 found
(c) A1 1/8 cao
(c) Answer: 1/8

Question 5

(a) M1 indices added: 1/2 + 3/2
(a) A1 x² cao
(a) Answer: x²
(b) M1 indices subtracted: 5 - (-2)
(b) A1 x⁷ cao
(b) Answer: x⁷
(c) M1 indices multiplied: (2/3) x 3
(c) A1 x² cao
(c) Answer: x²

Question 6

(a) M1 8^1/3 = 2 and x⁶ raised to power 1/3 gives x², both found
(a) A1 2x² cao
(a) Answer: 2x²
(b) M1 9^1/2 = 3 and x⁴ raised to power 1/2 gives x², both found
(b) A1 3x² cao
(b) Answer: 3x²
(c) M1 (x³)^-2 = x^-6 found
(c) A1 1/x² oe (x^-2)
(c) Answer: x^-2 (= 1/x²)

Question 7

(a) B1 x = 5 cao
(a) Answer: x = 5
(b) M1 1/9 written as 3^-2
(b) A1 x = -2 cao
(b) Answer: x = -2
(c) M1 1/√5 written as 5^-1/2
(c) A1 x = -1/2 oe
(c) Answer: x = -1/2

Question 8

(a) M1 both sides written as powers of 2: 2^2x = 2³
(a) A1 x = 3/2 oe
(a) Answer: x = 3/2
(b) M1 both sides written as powers of 3: 3^2x = 3^-1
(b) A1 x = -1/2 oe
(b) Answer: x = -1/2

Question 9

M1 (x⁴)^1/2 = x² found
M1 indices combined: 2 - (-3)
A1 x⁵ cao
Answer: x⁵

Question 10

B1 27^1/3 = 3 shown
B1 4^-1/2 = 1/2 shown
A1 3 * 1/2 = 3/2 shown, cso (dependent on both B marks)
Answer: 3/2 (shown)

Question 11

M1 √x³ written as x^3/2
M1 indices subtracted: 3/2 - 1/4
A1 x^5/4 oe (accept x^1.25)
Answer: x^5/4

Question 12

M1 25 written as 5²
M1 indices equated: 2x = 3x - 2
A1 x = 2 cao
Answer: x = 2

Question 13

M1 both sides written as powers of 2: 1/8 = 2^-3, 16 = 2⁴
M1 indices equated: -3x = 4
A1 x = -4/3 oe
Answer: x = -4/3

Question 14

(a) M1 4^-1/2 = 1/2 found
(a) A1 N = 400 cao
(a) Answer: N = 400
(b) M1 equation set up: 800 * 4^-t/2 = 100, or 4^-t/2 = 1/8
(b) M1 both sides written as powers of 2 and indices equated
(b) A1 t = 3 cao
(b) Answer: t = 3

Question 15

M1 8^x+1 written as 2^3x+3
M1 4^2x-1 written as 2^4x-2
dM1 indices equated and rearranged: 3x + 3 = 4x - 2 (dependent on both previous M marks)
A1 x = 5 cao
Answer: x = 5

Question 16

M1 16^3/4 = 8 found
M1 x⁸ raised to power 3/4 gives x⁶
A1 8x⁶ cao
Answer: 8x⁶

Question 17

M1 indices multiplied: (-2/3) x (-3)
A1 x² cao
Answer: x²

Question 18

M1 a = 3 and b = 5 identified
M1 a - b = -2 found
A1 1/4 oe (accept 0.25)
Answer: 1/4

Question 19

M1 9^x written as 3^2x, or (3^x)²
M1 3^x+1 written as 3 * 3^x
dM1 expression factorised to 3^x(3^x - 3) = 0 and 3^x = 0 rejected as impossible (dependent on both previous M marks)
A1 x = 1 cao
Answer: x = 1

Question 20

M1 (2ⁿ)^1/2 written as 2^n/2
M1 indices added: n/2 + (1-n)
A1 2^{1 - n/2} oe (accept 2^(2-n)/2)
Answer: 2^{1 - n/2}

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Fractional and Negative Indices: Fluency and Exam Drill - Worksheets, Questions and Revision

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6.2D Fractional and Negative Indices: Fluency and Exam Drill

Key Index Laws to Recall

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